1st Edition

Divided Spheres
Geodesics and the Orderly Subdivision of the Sphere

ISBN 9781466504295
Published July 30, 2012 by A K Peters/CRC Press
532 Pages

USD $220.00

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Book Description

This well-illustrated book—in color throughout—presents a thorough introduction to the mathematics of Buckminster Fuller’s invention of the geodesic dome, which paved the way for a flood of practical applications as diverse as weather forecasting and fish farms. The author explains the principles of spherical design and the three main categories of subdivision based on geometric solids (polyhedra). He illustrates how basic and advanced CAD techniques apply to spherical subdivision and covers modern applications in product design, engineering, science, games, and sports balls.

Table of Contents

Divided Spheres
Working with Spheres
Making a Point
An Arbitrary Number
Symmetry and Polyhedral Designs
Spherical Workbenches
Detailed Designs
Other Ways to Use Polyhedra
Additional Resources
Bucky’s Dome
Synergetic Geometry
Dymaxion Projection
Cahill and Waterman Projections
Vector Equilibrium
The First Dome
NC State and Skybreak Carolina
Ford Rotunda Dome
Marines in Raleigh
University Circuit
Kaiser’s Domes
Union Tank Car
Covering Every Angle
Additional Resources
Putting Spheres to Work
Tammes Problem
Spherical Viruses
Celestial Catalogs
Sudbury Neutrino Observatory
Climate Models and Weather Prediction
Honeycombs for Supercomputers
Fish Farming
Virtual Reality
Modeling Spheres
Dividing Golf Balls
Spherical Throwable Panoramic Camera
Hoberman’s MiniSphere
Rafiki’s Code World
Art and Expression
Additional Resources
Circular Reasoning
Lesser and Great Circles
Geodesic Subdivision
Circle Poles
Arc and Chord Factors
Where Are We?
Altitude-Azimuth Coordinates
Latitude and Longitude Coordinates
Spherical Trips
Separation Angle
Latitude Sailing
Spherical Coordinates
Cartesian Coordinates
ρ, φ, λ Coordinates
Spherical Polygons
Excess and Defect
Additional Resources
Distributing Points
Additional Resources
Polyhedral Frameworks
What Is a Polyhedron?
Platonic Solids
Archimedean Solids
Additional Resources
Golf Ball Dimples
Icosahedral Balls
Octahedral Balls
Tetrahedral Balls
Bilateral Symmetry
Subdivided Areas
Dimple Graphics
Additional Resources
Subdivision Schemas
Geodesic Notation
Triangulation Number
Frequency and Harmonics
Grid Symmetry
Class I: Alternates and Ford
Class II: Triacon
Class III: Skew
Covering the Whole Sphere
Additional Resources
Comparing Results
Sameness or Nearly So
Triangle Area
Face Acuteness
Euler Lines
Parts and T . 257
Convex Hull
Spherical Caps
Face Orientation
King Icosa
Additional Resources
Computer-Aided Design
A Short History
Octet Truss Connector
Spherical Design
Three Class II Triacon Designs
Panel Sphere
Class II Strut Sphere
Class II Parabolic Stellations
Class I Ford Shell
31 Great Circles
Class III Skew
Additional Resources
Advanced CAD Techniques
Reference Models
An Architectural Example
Spherical Reference Models
Prepackaged Reference and Assembly Models
Local Axis Systems
Assembly Review
Associative Geometry
Design-in-Context versus Constraints
Mirrored Enantiomorphs
Power Copy
Power Copy Prototype
Data Structures
CAD Alternatives: Stella and Antiprism
Additional Resources
Spherical Trigonometry
Basic Trigonometric Functions
The Core Theorems
Law of Cosines
Law of Sines
Right Triangles
Napier’s Rule
Using Napier’s Rule on Oblique Triangles
Polar Triangles
Additional Resources
Stereographic Projection
Points on a Sphere
Stereographic Properties
A History of Diverse Uses
The Astrolabe
Crystallography and Geology
Projection Methods
Great Circles
Lesser Circles
Wulff Net
Polyhedra Stereographics
Polyhedra as Crystals
Metrics and Interpretation
Projecting Polyhedra
Geodesic Stereographics
Spherical Icosahedron
Additional Resources
Geodesic Math
Class I: Alternates and Fords
Class II: Triacon
Class III: Skew
Characteristics of Triangles
Storing Grid Points
Additional Resources
Schema Coordinates
Coordinates for Class I: Alternates and Ford
Coordinates for Class II: Triacon
Coordinates for Class III: Skew
Coordinate Rotations
Rotation Concepts
Direction and Sequences
Simple Rotations
Antipodal Points
Compound Rotations
Rotation around an Arbitrary Axis
Polyhedra and Class Rotation Sequences
Icosahedron Classes I and III
Icosahedron Class
Octahedron Classes I and III
Octahedron Class
Tetrahedron Classes I and III
Tetrahedron Class
Dodecahedron Class
Cube Class
Implementing Rotations
Using Matrices
Rotation Algorithms
An Example
Additional Resources

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"… illustrations in the book, nearly all of them computer generated, are very good indeed. … The book contains an extremely detailed metrical treatment of all the regular and Archimedean polyhedra. An important construction is the space tessellating octahedron + tetrahedron which Fuller described as ‘simplest, most powerful structural system in the universe.’ Taking tubes along the edges of the tessellation, he devised and patented a joint to which up to nine tubes could be connected, making a very rigid structure. This is called the ‘octet struss connector’ and receives an entire, beautifully illustrated chapter in the book. … remarkable book … the sheer scale of the book, 509 pages on how to divide up the surface of a sphere, is amazing."
—Peter Giblin, The Mathematical Gazette, March 2014

"The text is written for designers, architects and people interesting in constructions of domes based on spherical subdivision. The book is illustrated with many figures and sketches and examples of real-life usage of the constructions developed during (roughly) the past 60 years. Overall, the book is written in a way accessible to a non-expert in mathematics and geometry. … The book could certainly be a good source for inspiration, with many applications, mostly in architecture and other related areas."
—Pavel Chalmoviansky, Mathematical Reviews, May 2013

"This well-illustrated book-in color throughout-presents a thorough introduction to the mathematics of Buckminster Fuller's invention of the geodesic dome, which paved the way for a flood of practical applications as diverse as weather forecasting and fish farms. The author explains the principles of spherical design and the three main categories of subdivision based on geometric solids (polyhedra). He illustrates how basic and advanced CAD techniques apply to spherical subdivision and covers modem applications in product design, engineering, science, games, and sports balls."

"… the ways in which spheres are modified that make them functional and more interesting … [are] the main point[s] of the book. … Implementations of tessellated spheres are used to describe real-world situations, from computer processor grids to fish farming to the surface of golf balls to global climate models. This is a very entertaining section, demonstrating once again how powerful and useful mathematics is. … this book is an existence proof of how complex, interesting and useful properly altered spheres can be."
—Charles Ashbacher, MAA Reviews, December 2012

"In support of his primer, Popko provides a glossary of over 300 terms, a bibliography of 385 citations, reference to 28 useful websites, and an index of nine double columned pages. For some readers, these aids will be most useful in accessing and keeping track of the great diversity of ideas and concepts as well as practical and analytical procedures found in this complex and engaging volume. … a broad array of readers will find much of interest and value in this volume whether in terms of mathematics, conceptualization, application, or production."
—Henry W. Castner, GEOMATICA, Vol. 66, No. 3, 2012

"I have loved the beauty and symmetry of polyhedra and spherical divisions for many years. My own efforts have been concentrated on making both simple and complex spherical models using classical methods and simple tools. Dr. Popko’s elegant new book extends both the science and the art of spherical modeling to include Computer-Aided Design and applications, which I would never have imagined when I started down this fascinating and rewarding path.
His lovely illustrations bring the subject to life for all readers, including those who are not drawn to the mathematics. This book demonstrates the scope, beauty and utility of an art and science with roots in antiquity. Spherical subdivision is relevant today and useful for the future. Anyone with an interest in the geometry of spheres, whether a professional engineer, an architect or product designer, a student, a teacher, or simply someone curious about the spectrum of topics to be found in this book, will find it helpful and rewarding."
—Magnus Wenninger, Benedictine Monk and Polyhedral Modeler

"Edward Popko’s Divided Spheres is the definitive source for the many varied ways a sphere can be divided and subdivided. From domes and pollen grains to golf balls, every category and type is elegantly described in these pages. The mathematics and the images together amount to a marvelous collection, one of those rare works that will be on the bookshelf of anyone with an interest in the wonders of geometry."
—Kenneth Snelson, Sculptor and Photographer

"Edward Popko’s Divided Spheres is a ‘thesaurus’ must to those whose academic interest in the world of geometry looks to greater coverage of synonyms and antonyms of this beautiful shape we call a sphere. The late Buckminster Fuller might well place this manuscript as an all-reference for illumination to one of nature’s most perfect invention."
—Thomas T.K. Zung, Senior Partner, Buckminster Fuller, Sadao & Zung Architects

"My own discovery, Waterman Polyhedra, was my way to see hidden patterns in ordered points in space. Ed's book Divided Spheres is about patterns and points too but on spheres. He shows you how to solve practical design problems based spherical polyhedra. Novices and experts will understand the challenges and classic techniques of spherical design just by looking at the many beautiful illustrations."
—Steve Waterman, Mathematician

"Ed Popko’s comprehensive survey of the history, literature, geometric and mathematical properties of the sphere is the definitive work on the subject. His masterful and thorough investigation of every aspect is covered with sensitivity and intelligence. This book should be in the library of anyone interested in the orderly subdivision of the sphere."
—Shoji Sadao, Architect, Cartographer, and Lifelong Business Partner of Buckminster Fuller

"Any math collection concerned with spherical modeling will find this offers a basic yet complex introduction … blends art with scientific inquiry, providing a college-level coverage of geometry that will bring math alive for any who want a discussion of sphere science."
—Midwest Book Review